I went out for the first ride in ages on Suday, and my LX shifter for the rear mech was playing up. Pushing the thumb lever to go through the gears, you normally get one gear with a light push, or you can go through the lever's full travel to get two or three (conventional rear mech setup, so this is changing down into lower gears).

However, on Sunday the lever had to be pushed right to the end of its travel until it would click through a gear. Any ideas on how to remedy it?

Assuming that the cables are all running well I'd be tempted to take the cover off the top of the shifter and blast it with WD40 or similar. It may be something simple like a bit of muck getting in the mechanism. I have also found my Shimano shifter on my commuter to be occasionally temperamental with the cold.

strip it down and clean all the mechanisms with WD40, then rebuild….